TSTP Solution File: SEV038^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV038^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n111.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:37 EDT 2014
% Result : Timeout 300.02s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV038^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n111.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:39:36 CDT 2014
% % CPUTime : 300.02
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1b396c8>, <kernel.Type object at 0x1b39fc8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula ((ex (((a->Prop)->Prop)->(a->(a->Prop)))) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) of role conjecture named cTHM266_pme
% Conjecture to prove = ((ex (((a->Prop)->Prop)->(a->(a->Prop)))) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((ex (((a->Prop)->Prop)->(a->(a->Prop)))) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U))))))))']
% Parameter a:Type.
% Trying to prove ((ex (((a->Prop)->Prop)->(a->(a->Prop)))) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U))))))))
% Found eq_ref00:=(eq_ref0 (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))):(((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U))))))))
% Found (eq_ref0 (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) b)
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) b)
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) b)
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U))))))))
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U))))))))
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U))))))))
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) b) (fun (F:(((a->Prop)->Prop)->(a->(a->Prop))))=> ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((F P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((F P) Xx) Xy)->(((F P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((F P) Xx) Xy)) (((F P) Xy) Xz))->(((F P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (F P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (F T)) (F U))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) b)
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) b)
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) b)
% Found ((eq_ref ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) as proof of (((eq ((((a->Prop)->Prop)->(a->(a->Prop)))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))):(((eq Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))
% Found (eq_ref0 (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) b)
% Found ((eq_ref Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) b)
% Found ((eq_ref Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) b)
% Found ((eq_ref Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found (((eq_trans000 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((((eq_trans00 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found (((eq_trans000 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((((eq_trans00 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found eq_ref00:=(eq_ref0 (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))):(((eq Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))
% Found (eq_ref0 (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) as proof of (((eq Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) b)
% Found ((eq_ref Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) as proof of (((eq Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) b)
% Found ((eq_ref Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) as proof of (((eq Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) b)
% Found ((eq_ref Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) as proof of (((eq Prop) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))))) b)
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found (((eq_sym0 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and 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(a->(a->Prop))) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found eq_ref00:=(eq_ref0 (x x00)):(((eq (a->(a->Prop))) (x x00)) (x x00))
% Found (eq_ref0 (x x00)) as proof of (((eq (a->(a->Prop))) (x x00)) R)
% Found ((eq_ref (a->(a->Prop))) (x x00)) as proof of (((eq (a->(a->Prop))) (x x00)) R)
% Found ((eq_ref (a->(a->Prop))) (x x00)) as proof of (((eq (a->(a->Prop))) (x x00)) R)
% Found ((eq_ref (a->(a->Prop))) (x x00)) as proof of (((eq (a->(a->Prop))) (x x00)) R)
% Found (eq_sym000 ((eq_ref (a->(a->Prop))) (x x00))) as proof of (((eq (a->(a->Prop))) R) (x x00))
% Found ((eq_sym00 R) ((eq_ref (a->(a->Prop))) (x x00))) as proof of (((eq (a->(a->Prop))) R) (x x00))
% Found (((eq_sym0 (x x00)) R) ((eq_ref (a->(a->Prop))) (x x00))) as proof of (((eq (a->(a->Prop))) R) (x x00))
% Found ((((eq_sym (a->(a->Prop))) (x x00)) R) ((eq_ref (a->(a->Prop))) (x x00))) as proof of (((eq (a->(a->Prop))) R) (x x00))
% Found ((((eq_sym (a->(a->Prop))) (x x00)) R) ((eq_ref (a->(a->Prop))) (x x00))) as proof of (((eq (a->(a->Prop))) R) (x x00))
% Found x000:(x00 Xp)
% Found x000 as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (x000:(x00 Xp))=> x000) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (Xp:(a->Prop)) (x000:(x00 Xp))=> x000) as proof of ((x00 Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))
% Found (fun (Xp:(a->Prop)) (x000:(x00 Xp))=> x000) as proof of (forall (Xp:(a->Prop)), ((x00 Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found x000:(x00 Xp)
% Found x000 as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (x000:(x00 Xp))=> x000) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (Xp:(a->Prop)) (x000:(x00 Xp))=> x000) as proof of ((x00 Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))
% Found (fun (Xp:(a->Prop)) (x000:(x00 Xp))=> x000) as proof of (forall (Xp:(a->Prop)), ((x00 Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz)))) b)
% Found eq_ref00:=(eq_ref0 (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P)))))
% Found (eq_ref0 (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found eta_expansion000:=(eta_expansion00 T):(((eq ((a->Prop)->Prop)) T) (fun (x:(a->Prop))=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 T):(((eq ((a->Prop)->Prop)) T) (fun (x:(a->Prop))=> (T x)))
% Found (eta_expansion_dep00 T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found eta_expansion000:=(eta_expansion00 T):(((eq ((a->Prop)->Prop)) T) (fun (x:(a->Prop))=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found eta_expansion_dep000:=(eta_expansion_dep00 T):(((eq ((a->Prop)->Prop)) T) (fun (x:(a->Prop))=> (T x)))
% Found (eta_expansion_dep00 T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) U)
% Found eta_expansion000:=(eta_expansion00 T):(((eq ((a->Prop)->Prop)) T) (fun (x:(a->Prop))=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found (((eta_expansion (a->Prop)) Prop) T) as proof of (((eq ((a->Prop)->Prop)) T) b)
% Found x000:=(x00 (fun (x9:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x00 (fun (x9:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x00 (fun (x9:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x080:=(x08 (fun (x1:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x08 (fun (x1:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x08 (fun (x1:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x020:=(x02 (fun (x7:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x02 (fun (x7:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x02 (fun (x7:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x040:=(x04 (fun (x5:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x04 (fun (x5:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x04 (fun (x5:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found (((eq_trans000 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found ((((eq_trans00 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found (((eq_trans000 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found ((((eq_trans00 ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((and ((and (forall (Xx:a), (((x P) Xx) Xx))) (forall (Xx:a) (Xy:a), ((((x P) Xx) Xy)->(((x P) Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and (((x P) Xx) Xy)) (((x P) Xy) Xz))->(((x P) Xx) Xz))))))) (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((ex ((a->Prop)->Prop)) (fun (P:((a->Prop)->Prop))=> ((and ((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (((eq (a->(a->Prop))) R) (x P))))))))) (forall (T:((a->Prop)->Prop)) (U:((a->Prop)->Prop)), (((and ((and ((and ((and (not (((eq ((a->Prop)->Prop)) T) U))) (forall (Xp:(a->Prop)), ((T Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (T Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (T Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))) (forall (Xp:(a->Prop)), ((U Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (U Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (U Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->(not (((eq (a->(a->Prop))) (x T)) (x U)))))))))
% Found x060:=(x06 (fun (x3:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x06 (fun (x3:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x06 (fun (x3:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x000:=(x00 (fun (x9:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x00 (fun (x9:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x00 (fun (x9:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x000:=(x00 (fun (x9:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x00 (fun (x9:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x00 (fun (x9:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x080:=(x08 (fun (x1:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x08 (fun (x1:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x08 (fun (x1:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x080:=(x08 (fun (x1:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x08 (fun (x1:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found (x08 (fun (x1:(a->(a->Prop)))=> (P T))) as proof of (P0 T)
% Found x020:=(x02 (fun (x7:(a->(a->Prop)))=> (P T))):((P T)->(P T))
% Found (x02 (fu
% EOF
%------------------------------------------------------------------------------